A&A 624, A41 (2019) Astronomy https://doi.org/10.1051/0004-6361/201834962 & © ESO 2019 Astrophysics Analytical ray-tracing in planetary atmospheres A. Bourgoin, M. Zannoni, and P. Tortora Dipartimento di Ingegneria Industriale, University of Bologna, Via fontanelle 40, Forlì, Italy e-mail: [email protected] Received 21 December 2018 / Accepted 2 February 2019 ABSTRACT Context. Ground-based astro-geodetic observations and atmospheric radio occultations are two examples of observational techniques requiring a scrutiny analysis of atmospheric refraction. In both cases, the measured changes of the observables are geometrically related to changes in the photon path through the refractive profile of the crossed medium. Therefore, having a clear knowledge of how the refractivity governs the photon path evolution is of prime importance to clearly understand observational features. Aims. We analytically performed the integration of the photon path and the light time of rays traveling across a non-spherically symmetric planetary atmosphere. Methods. Assuming that the atmospheric refraction evolves linearly with the Newtonian potential, we derived an exact solution to the equations of geometrical optics. By varying the solution’s arbitrary constants of integration, we reformulated the equation of geometrical optics into a new set of osculating equations describing the constants’ evolution following any changes in the refractive profile. We have highlighted the capabilities of the formalism, carrying out five realistic applications in which we derived analytical expressions. Finally, we assessed the accuracy by comparing the solution to results from a numerical integration of the equations of geometrical optics in the presence of a quadrupolar moment (J2). Results. Analytical expressions for the light time and the refractive bending are given with relative errors at the level of one part in 8 5 4 2 10 and one part in 10 , for typical values of the refractivity and J2 at levels of 10− and 10− , respectively. Conclusions. The establishment of the osculating equations for the ray propagation has two main advantages. Firstly, it provides an easy and comprehensive geometrical picture for interpreting the photon path. Secondly, it allows the analytical solving of the ray propagation in the presence of non-radial dependencies in the refractive profile. Key words. atmospheric effects – methods: analytical – occultations – planets and satellites: atmospheres 1. Introduction usually occurs at characteristic lengths which are much larger than the wavelength of the electromagnetic signal and the prob- In the context of Maxwell’s theory of electromagnetism, the lem can be studied in the approximation of geometrical optics. In refraction phenomenon is understood as the superposition of that context, the different observables (range, Doppler, or atten- an incident electromagnetic wave with wavelets generated by uation) can be related to the geometry of the photon path and some electric particles being accelerated in response to a local the light time which are both controlled by the index of refrac- electromagnetic field. If the medium containing the particles is tion. Thus, a clear knowledge of how the geometry of the light sufficiently tenuous (as for a gas), the local field is proportional propagation is related to the index of refraction is important to to the incident field itself. The waves’ superposition generates understand the changes in observables. a resulting signal which exhibits a phase shift with respect to For GB observations, especially for techniques operating for the incident one. This apparent change in the primary signal’s the realization of the international Earth rotation and reference phase is parametrized thanks to a dimensionless parameter called systems service (IERS), the accuracy of the analysis is largely the index of refraction of the medium. It encloses the physical affected by errors in modeling the group delay during propa- properties related to the interaction between the electromagnetic gation of the signal through the atmosphere (Petit et al. 2010). signal and the material filling the medium. Therefore, any vari- The atmospheric delay is usually evaluated at zenith and is com- ation of the physical properties of the medium will change the puted with an analytical model of Marini & Murray(1973) or index of refraction which in turn will modify the features of with the more recent model (also analytic) of Mendes & Pavlis the incident signal. In Astronomy, those changes in the signal’s (2004). Then, the projection to a given elevation angle is oper- characteristics are the observables, thus, it is of prime impor- ated with the help of the mapping function developed by Mendes tance to understand the relationship between refraction and light et al.(2002). Those models are formulated under the assumption propagation. that the atmosphere is spherically symmetric and assume a radial Ground-based (GB) astro-geodetic observations, atmo- dependency for the index of refraction. Consequently, they leave spheric radio occultations (ARO), and atmospheric stellar occul- aside some possible effects due to the presence of horizontal tations (ASO) are three examples of observational techniques gradients in the atmosphere, which might be somehow inaccu- requiring a careful analysis of atmospheric refraction. For GB rate. (These gradients could arise at two different levels. Firstly, observations, refraction is an accuracy-limiting factor and it they could be generated by an atmosphere with a non-spherical has to be modeled in the analysis software; for both ARO and shape. Secondly, they could result from a local horizontal ASO, it is thoroughly modeled to infer the physical properties variation of the physical quantities entering in the computation of the refractive medium. For all these techniques, the refraction of the index of refraction such as the temperature.) This issue has Article published by EDP Sciences A41, page 1 of 30 A&A 624, A41 (2019) only been partially overcome by Chen & Herring(1997) for very observables, which are related to the light geometry, can also be long baseline interferometry (VLBI), and by Hulley & Pavlis expressed in terms of this parameter. In this paper, we present (2007) for satellite laser ranging (SLR), by performing numerical a useful mathematical framework which allows to perform ana- integration (numerical ray-tracing) across multi-layered spheri- lytical studies beyond the usual spherically symmetric case. The cal atmosphere, assuming constant refractivity inside each layer. validity of the derived solution is assessed by comparing it with Even if in those cases the index of refraction may possess numerical ray-tracing performed on a simulated atmosphere. We non-radial dependencies, the shape of the atmosphere is still try to keep the discussion as general as possible in order to let the considered spherical. It is also worth mentioning that beyond possibility of applying the incoming description to any situation the radial dependency of the refractive index, only numerical involving light propagation in planetary atmospheres such as GB integrations are usually used for such GB observations. observations, ARO or ASO experiments. In ARO and ASO experiments, the basic idea is to estab- lish electromagnetic links between a transmitter and a receiver when the former is being occulted by a planetary atmosphere 2. Definitions and hypothesis as seen from the receiver. (For ARO experiments, a radio link In this paper, we use the geometrical optics approximation which is established, while for ASO experiments the transmitter is a is characterized by neglecting the wavelength of electromag- distant star, so that the signal’s wavelengths usually range from netic waves (Landau & Lifshitz 1975). In geometrical optics, the near infrared to visible. In the following, we mainly focus on concept of rays is introduced as curves whose tangents at each ARO experiments, but the entire discussion applies also to ASO point coincide with the direction of propagation of the electro- experiments.) Conversely to GB observations, ARO experiments magnetic waves. In the following, we will mainly focus on the use the effect of the refraction by the atmosphere to determine evolution of the separation vector, locating a point along the light the physical properties of the medium. This is often referred to ray, and the unit vector tangent to the ray at that point. In this as the inverse problem. In the literature, there exists different section, we introduce the basic principles for determining the approaches devoted to the resolution of the inverse problem. For evolution of these quantities. We provide in Table A.1 a list of instance, the initial data processing of the Mariner IV occulta- the nomenclature used hereafter. tions was based primarily on model fitting (Kliore et al. 1965). The model used for determining the index of refraction pro- file was approximate and only valid for a planet with a thin 2.1. Equations of geometrical optics spherically symmetric atmosphere (Fjeldbo & Eshleman 1965, Let us introduce h, the separation vector between the center of a 1968). Later, it has been shown (Phinney & Anderson 1968) that reference frame and a point along the light ray trajectory. Also, determining directly the refractivity profile from the Doppler fre- we define s, the unit vector which is tangent to the light ray at quencies (or from the bending angle) is a special case of Abelian h’s location integral inversion when the index